scholarly journals Extension of Castigliano’s method for isotropic beams

2020 ◽  
Vol 231 (11) ◽  
pp. 4621-4640
Author(s):  
Juergen Schoeftner
Keyword(s):  
Shear Force ◽  
Strain Energy ◽  
Axial Stress ◽  
Bending Moment ◽  
Axial Direction ◽  
Present Contribution ◽  
Special Cases ◽  
Dummy Load ◽  
Complementary Strain ◽  
Castigliano’S Theorem

Abstract In the present contribution Castigliano’s theorem is extended to find more accurate results for the deflection curves of beam-type structures. The notion extension in the context of the second Castigliano’s theorem means that all stress components are included for the computation of the complementary strain energy, and not only the dominant axial stress and the shear stress. The derivation shows that the partial derivative of the complementary strain energy with respect to a scalar dummy parameter is equal to the displacement field multiplied by the normalized traction vector caused by the dummy load distribution. Knowing the Airy stress function of an isotropic beam as a function of the bending moment, the normal force, the shear force and the axial and vertical load distributions, higher-order formulae for the deflection curves and the cross section rotation are obtained. The analytical results for statically determinate and indeterminate beams for various load cases are validated by analytical and finite element results. Furthermore, the results of the extended Castigliano theory (ECT) are compared to Bernoulli–Euler and Timoshenko results, which are special cases of ECT, if only the energies caused by the bending moment and the shear force are considered. It is shown that lower-order terms for the vertical deflection exist that yield more accurate results than the Timoshenko theory. Additionally, it is shown that a distributed load is responsible for shrinking or elongation in the axial direction.

Structural Topology Optimization Using Frame Elements Based on the Complementary Strain Energy Concept for Eigen-Frequency Maximization

2005 ◽  
Author(s):  
Akihiro Takezawa ◽  
Shinji Nishiwaki ◽  
Kazuhiro Izui ◽  
Masataka Yoshimura
Keyword(s):  
Topology Optimization ◽  
Cross Sections ◽  
Strain Energy ◽  
Optimization Procedure ◽  
Optimization Method ◽  
Structural Topology ◽  
Energy Concept ◽  
Conceptual Design Phase ◽  
Complementary Strain ◽  
Topology Optimization Method

This paper discuses a new topology optimization method using frame elements for the design of mechanical structures at the conceptual design phase. The optimal configurations are determined by maximizing multiple eigen-frequencies in order to obtain the most stable structures for dynamic problems. The optimization problem is formulated using frame elements having ellipsoidal cross-sections, as the simplest case. Construction of the optimization procedure is based on CONLIN and the complementary strain energy concept. Finally, several examples are presented to confirm that the proposed method is useful for the topology optimization method discussed here.

The Behaviour of Multiple Loops in Torsion

1988 ◽  
Vol 15 (3) ◽  
pp. 149-156 ◽  
Author(s):  
R. A. Cavina ◽  
N. E. Waters
Keyword(s):  
Energy Method ◽  
Strain Energy ◽  
Vertical Axis ◽  
Experimental Measurements ◽  
Complementary Strain ◽  
Radial Axis ◽  
Strain Energy Method ◽  
Good Agreement

The angular stiffness of a multiple looped span, subject to rotation about a vertical axis (torsion) and also to rotation about a horizontal or radial axis (mesio-distal tilt), have been derived using the complementary (strain) energy method. Experimental measurements on enlarged models were in good agreement with the values calculated from the theoretical relationships obtained. The variations in angular stiffness resulting from changes in the loop height, width, and position of clinical sized loops are discussed.

Crack Driving Force in Anisotropic Media due to Non-Elastic Deformation

2004 ◽  
Vol 261-263 ◽  
pp. 75-80
Author(s):  
G.H. Nie ◽  
H. Xu
Keyword(s):  
Elastic Deformation ◽  
Driving Force ◽  
Strain Energy ◽  
Elastic Stress ◽  
Complex Function ◽  
Crack Driving Force ◽  
Special Cases ◽  
Degree Of Anisotropy ◽  
Minimum Strain Energy ◽  
Orthotropic Media

In this paper elastic stress field in an elliptic inhomogeneity embedded in orthotropic media due to non-elastic deformation is determined by the complex function method and the principle of minimum strain energy. Two complex parameters are expressed in a general form, which covers all characterizations of the degree of anisotropy for any ideal orthotropic elastic body. The stress acting on the long side of ellipse can be considered as a crack driving force and applied in failure and fatigue analysis of composites. For some special cases, the resulting solutions will reduce to the known results.

Ratcheting assessment of corroded elbow pipes subjected to internal pressure and cyclic bending moment

2021 ◽  
pp. 030932472198989
Author(s):  
Ali Salehi ◽  
Armin Rahmatfam ◽  
Mohammad Zehsaz
Keyword(s):  
Growth Rate ◽  
Stainless Steel ◽  
Internal Pressure ◽  
Bending Moment ◽  
Axial Direction ◽  
Hoop Strain ◽  
Cyclic Bending ◽  
High Fatigue ◽  
The Impact ◽  
Cyclic Bending Moment

The present study aimed to study ratcheting strains of corroded stainless steel 304LN elbow pipes subjected to internal pressure and cyclic bending moment. To this aim, spherical and cubical shapes corrosion are applied at two depths of 1 mm and 2 mm in the critical points of elbow pipe such as symmetry sites at intrados, extrados, and crown positions. Then, a Duplex 2205 stainless steel elbow pipe is considered as an alternative to studying the impact of the pipe materials, due to its high corrosion resistance and strength, toughness, and most importantly, the high fatigue strength and other mechanical properties than stainless steel 304LN. In order to perform numerical analyzes, the hardening coefficients of the materials were calculated. The results highlight a significant relationship between the destructive effects of corrosion and the depth and shape of corrosion, so that as corrosion increases, the resulting destructive effects increases as well, also, the ratcheting strains in cubic corrosions have a higher growth rate than spherical corrosions. In addition, the growth rate of the ratcheting strains in the hoop direction is much higher across the studied sample than the axial direction. The highest growth rate of hoop strain was observed at crown and the highest growth rate of axial strains occurred at intrados position. Altogether, Duplex 2205 material has a better performance than SS 304LN.

Imperialist Competitive Algorithm for Multiobjective Optimization of Ellipsoidal Head of Pressure Vessel

2011 ◽  
Vol 110-116 ◽  
pp. 3422-3428 ◽  
Author(s):  
Behzad Abdi ◽  
Hamid Mozafari ◽  
Ayob Amran ◽  
Roya Kohandel
Keyword(s):  
Genetic Algorithm ◽  
Pressure Vessel ◽  
Shear Force ◽  
Minimum Weight ◽  
Bending Moment ◽  
Imperialist Competitive Algorithm ◽  
Optimum Shape ◽  
Working Pressure ◽  
Competitive Algorithm ◽  
Bending Stresses

This work devoted to an ellipsoidal head of pressure vessel under internal pressure load. The analysis is aimed at finding an optimum weight of ellipsoidal head of pressure vessel due to maximum working pressure that ensures its full charge with stresses by using imperialist competitive algorithm and genetic algorithm. In head of pressure vessel the region of its joint with the cylindrical shell is loaded with shear force and bending moments. The load causes high bending stresses in the region of the joint. Therefore, imperialist competitive algorithm was used here to find the optimum shape of a head with minimum weight and maximum working pressure which the shear force and the bending moment moved toward zero. Two different size ellipsoidal head examples are selected and studied. The imperialist competitive algorithm results are compared with the genetic algorithm results.

Elastic-plastic bending of the pipeline under combined loading

2021 ◽  
pp. 372-377
Author(s):  
Виктор Миронович Варшицкий ◽  
Евгений Павлович Студёнов ◽  
Олег Александрович Козырев ◽  
Эльдар Намикович Фигаров
Keyword(s):  
Internal Pressure ◽  
Limit State ◽  
Bending Moment ◽  
Axial Direction ◽  
Longitudinal Force ◽  
Combined Loading ◽  
Dimensional Problem ◽  
Elastic Plastic ◽  
Pipeline Section ◽  
Thin Walled Pipe

Рассмотрена задача упругопластического деформирования тонкостенной трубы при комбинированном нагружении изгибающим моментом, осевой силой и внутренним давлением. Решение задачи осуществлено по разработанной методике с помощью математического пакета Matcad численным методом, основанным на деформационной теории пластичности и безмоментной теории оболочек. Для упрощения решения предложено сведение двумерной задачи к одномерной задаче о деформировании балки, материал которой имеет различные диаграммы деформирования при сжатии и растяжении в осевом направлении. Проведено сравнение с результатами численного решения двумерной задачи методом конечных элементов в упругопластической постановке. Результаты расчета по инженерной методике совпадают с точным решением с точностью, необходимой для практического применения. Полученные результаты упругопластического решения для изгибающего момента в сечении трубопровода при комбинированном нагружении позволяют уточнить известное критериальное соотношение прочности сечения трубопровода с кольцевым дефектом в сторону снижения перебраковки. Применение разработанной методики позволяет ранжировать участки трубопровода с непроектным изгибом по степени близости к предельному состоянию при комбинированном нагружении изгибающим моментом, продольным усилием и внутренним давлением. The problem of elastic plastic deformation of a thin-walled pipe under co-binned loading by bending moment, axial force and internal pressure is considered. The problem is solved by the developed method using the Matcad mathematical package by a numerical method based on the deformation theory of plasticity and the momentless theory of shells. To simplify the solution of the problem, it is proposed to reduce a twodimensional problem to a one-dimensional problem about beam deformation, the material of which has different deformation diagrams under compression and tension in the axial direction. Comparison with the results of numerical solution of the two-dimensional problem with the finite element method in the elastic plastic formulation is carried out. The obtained results of the elastic-plastic solution for the bending moment in the pipeline section under combined loading make it possible to clarify criterion ratio of the strength of the pipeline section with an annular defect in the direction of reducing the rejection. Application of the developed approach allows to rank pipeline sections with non-design bending in the steppe close to the limit state under combined loading of the pipeline with bending moment, longitudinal force and internal pressure.

Bending of Thin Ring-Sector Plates

1951 ◽  
Vol 18 (4) ◽  
pp. 359-363
Author(s):  
L. I. Deverall ◽  
C. J. Thorne
Keyword(s):  
Continuous Function ◽  
Bending Moment ◽  
Circular Ring ◽  
Circular Plates ◽  
Thin Ring ◽  
Special Cases ◽  
Edge Conditions

Abstract General expressions for the deflection of plates whose planform is a sector of a circular ring are given for cases in which the straight edges have arbitrary but given deflection and bending moment. The solutions are given for all combinations of physically important edge conditions on the two circular edges. Sectors of circular plates are included as special cases. Solutions are given for a general load which is a continuous function of r, and a sectionally continuous function of θ, where r and θ are the usual polar co-ordinates with the pole at the center of the ring. Several specific examples are given.

Application of the Mode-Acceleration Technique to the Solution of the Moving Oscillator Problem

1999 ◽  
Author(s):  
Alexander V. Pesterev ◽  
Lawrence A. Bergman
Keyword(s):  
Shear Force ◽  
Series Representation ◽  
Bending Moment ◽  
Static Solution ◽  
Distributed Parameter ◽  
Distributed Parameter System ◽  
Parameter System ◽  
One Dimensional ◽  
Acceleration Technique ◽  
Moving Oscillator

Abstract The problem of calculating the dynamic response of a one-dimensional distributed parameter system excited by an oscillator traversing the system with an arbitrarily varying speed is investigated. An improved series representation for the solution is derived that takes into account the jump in the shear force at the point of the attachment of the oscillator, which makes it possible to efficiently calculate the distributed shear force and, where applicable, bending moment. The improvement is achieved through the introduction of the “quasi-static” solution, an approximation to the desired one, which makes it possible to apply to the moving oscillator problem the “mode-acceleration” technique conventionally used for acceleration of series in problems related to the steady-state vibration of distributed systems. Numerical results illustrating the efficiency of the method are presented.

The Influence of Rotatory Inertia and Transverse Shear on the Dynamic Plastic Behavior of Beams

1979 ◽  
Vol 46 (2) ◽  
pp. 303-310 ◽  
Author(s):  
Norman Jones ◽  
J. Gomes de Oliveira
Keyword(s):  
Transverse Shear ◽  
Shear Force ◽  
Yield Criterion ◽  
Yield Condition ◽  
Bending Moment ◽  
Plastic Behavior ◽  
Plastic Response ◽  
Rotatory Inertia ◽  
Perfectly Plastic ◽  
Theoretical Procedure

The theoretical procedure presented herein examines the influence of retaining the transverse shear force in the yield criterion and rotatory inertia on the dynamic plastic response of beams. Exact theoretical rigid perfectly plastic solutions are presented for a long beam impacted by a mass and a simply supported beam loaded impulsively. It transpires that rotatory inertia might play a small, but not negligible, role on the response of these beams. The results in the various figures indicate that the greatest departure from an analysis which neglects rotatory inertia but retains the influence of the bending moment and transverse shear force in the yield condition is approximately 11 percent for the particular range of parameters considered.

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